The Conjugacy Problem and Higman Embeddings

For every finitely generated recursively presented group $mathcal G$ we construct a finitely presented group $mathcal H$ containing $mathcal G$ such that $mathcal G$ is (Frattini) embedded into $mathcal H$ and the group $mathcal H$ has solvable conjugacy problem if and only if $mathcal G$ has solvable conjugacy problem.